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Last Updated: 04.04.2023
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Fig. 1: Stay here and learn more to keep your brain from getting the metaphorical bends!
Test your knowledge with multiple choice flashcards
Hydrostatic Pressure Definition
So what is hydrostatic pressure?
Hydrostatic pressure
is the pressure exerted by a fluid at rest due to the force of gravity.
Hydrostatic pressure varies depending on the fluid and the depth.
Hydrostatic Pressure Formula
You will need some variables going forward:
P
is the hydrostatic pressure in
N
/
m
2
;
ρ
is the density of the fluid in
kg
/
m
3
;
g
is the acceleration due to gravity in
m
/
s
2
; and
h
is the height of the water column in
m
.
Then the formula for hydrostatic pressure is
P
=
ρ
g
h
.
Notice that pressure increases as the height of the water column (also called the depth) increases. That fact is illustrated in the image below.
Fig. 2: The hydrostatic pressure, ranked from lowest to highest, would go
A
,
B
,
C
.
The density of water is
1000
kg
/
m
3
, and for gravity on Earth you can use
9.81
m
/
s
2
.
Hydrostatic Force
You are probably wondering what the difference between hydrostatic force and pressure is.
Hydrostatic force
is the force caused by the pressure of a fluid acting on a submerged surface.
If you think about the example of the diver at the start of the article, the hydrostatic force on the diver is the force of the water on their submerged body. As you have seen, pressure increases as the height of the water column increases. In other words, the deeper the diver goes, the higher the pressure is, which in turn means the hydrostatic force acting on their body increases as well.
Hydrostatic Force Formula
The hydrostatic force formula is
F
=
P
A
,
where:
F
is the hydrostatic force in
N
;
P
is they hydrostatic pressure in
N
/
m
2
; and
A
is the area of the submerged surface in
m
2
.
Notice that the formula works for very complicated surfaces, like that of a diver's body. However figuring out the exact hydrostatic force on a diver is a bit of a big problem to start with, so instead let's look at submerging a flat plate in a fluid instead.
Fig. 3: A flat plate submerged
d
units under the water.
In the picture above, there is a flat hexagonal plate submerged
d
units under the water. The rectangle shown has height
Δ
y
and width
x
i
, so the area of the rectangle is
x
i
Δ
y
. That means the hydrostatic force on the rectangle is
F
rectangle
=
P
A
=
(
ρ
g
h
)
(
x
i
Δ
y
)
=
(
1000
⋅
9.81
d
i
)
(
x
i
Δ
y
)
=
9810
d
i
x
i
Δ
y
N
.
An approximation of the hydrostatic force on the entire plate would be the sum of the hydrostatic forces on all of the rectangles, so if there are
n
rectangles,
F
plate
≈
∑
i
=
i
n
9810
d
i
x
i
Δ
y
.
This should look familiar to you since finding areas this way is one of the applications of integrals!
Hydrostatic Forces on Submerged Surfaces
Let's continue on with the plate example started in the previous section. The first step is to make a properly labeled diagram with the
x
and
y
-axis included. This plate is vertically symmetric, so it makes sense to center it on the
y
-axis with the bottom of the plate on the
x
axis.
The top of the plate is
6
meters below the surface of the water. The plate can be thought of in two sections: the upper section where
y
≥
2
; and the lower section where
y
≤
2
.
Fig. 4: The plate placed on the coordinate axis with the depth and equation of the upper line of the plate labeled.
Then the total area of the plate is the sum of the upper section area and the lower section area. Therefore when integrating you can split the integral at
y
=
2
to make calculations easier. In fact, since the area is symmetric across the
y
axis, you can find the area of the right half of the plate and then multiply by
2
to get the total area of the plate.
Upper Plate Calculations
For the upper part of the plate, the depth below the surface is
d
i
. The equation of the line describing the edge of the plate is
y
=
−
0.8
x
+
8
.
The summation for the hydrostatic force on the upper part of the plate is
F
upper
≈
∑
i
=
i
n
9810
d
i
x
i
Δ
y
,
which has a
Δ
y
in it. This means the integration will be done with respect to
y
and not with respect to
x
. Therefore solving the equation of the line for
x
in terms of
y
gives
x
=
−
1.25
y
+
10.
You will need to be careful since writing it this way only gives you the right half of the plate! Since the plate is symmetric, to find the area from the left side of the plate to the right side of the plate, you can multiply the summation by
2
. Substituting the line equation into the summation and multiplying by
2
gives
F
upper
≈
2
∑
i
=
i
n
9810
d
i
(
−
1.25
y
i
+
10
)
Δ
y
.
That just leaves writing
d
i
in terms of
y
i
. As you can see from the picture above,
y
i
=
12
−
d
i
,
and now the summation becomes
F
upper
≈
2
∑
i
=
i
n
9810
(
12
−
y
i
)
(
−
1.25
y
i
+
10
)
Δ
y
.
Then the total force on the upper half of the plate is
F
upper
=
2
lim
n
→
∞
∑
i
=
1
n
9810
(
12
−
y
i
)
(
−
1.25
y
i
+
10
)
Δ
y
=
2
∫
2
6
9810
(
12
−
y
)
(
−
1.25
y
+
10
)
d
y
=
2
(
9810
)
(
5
y
3
−
150
y
2
+
1440
y
12
)
|
y
=
2
y
=
6
=
2
(
9810
)
(
500
3
)
=
3270000
N
.
Lower Plate Calculations
Let's look at the lower plate. The diagram below shows the equation of the line for that part of the plate.
Fig. 5: Lower half of the submerged plate.
The equation of the line is
y
=
0.4
x
−
1
. As in the calculations for the upper half of the plate, you need to write this equation in terms of
x
so it can be substituted in. Rewriting the equation of the line in terms of
x
you get
x
=
2.5
y
+
2.5
.
Since
d
i
=
12
−
y
i
,
the hydrostatic force on a horizontal rectangular strip running from the
y
-axis to the line is
F
i
=
P
i
A
i
=
9810
d
i
(
2.5
y
i
+
2.5
)
Δ
y
=
9810
(
12
−
y
i
)
(
2.5
y
i
+
2.5
)
Δ
y
.
The total force on the upper half of the plate is then
F
lower
=
2
lim
n
→
∞
∑
i
=
1
n
9810
(
12
−
y
i
)
(
2.5
y
i
+
2.5
)
Δ
y
=
2
(
2.5
)
(
9810
)
∫
0
2
(
12
−
y
)
(
y
+
1
)
d
y
=
(
49050
)
−
2
y
3
+
33
y
2
+
72
y
6
|
y
=
0
y
=
2
=
(
49050
)
(
130
3
)
=
2125500
N
.
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
Total Hydrostatic Force on the Plate
Then to get the total hydrostatic force on the plate, you just need to add the hydrostatic force on the upper part of the plate to the
hydrostatic force on the lower part of the plate:
F
total
=
F
upper
+
F
lower
=
3270000
+
2125500
=
5395500
N
.
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
Hydrostatic Pressure - Key takeaways
Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity.
Then the formula for hydrostatic pressure is
P
=
ρ
g
h
.
where:
P
is the hydrostatic pressure in
N
/
m
2
;
ρ
is the density of the fluid in
kg
/
m
3
;
g
is the acceleration due to gravity in
m
/
s
2
; and
h
is the height of the water column in
m
.
Hydrostatic force is the force caused by the pressure of a fluid acting on a submerged surface.
The hydrostatic force formula is
F
=
P
A
,
where:
F
is the hydrostatic force in
N
;
P
is they hydrostatic pressure in
N
/
m
2
; and
A
is the area of the submerged surface in
m
2
.
Frequently Asked Questions about Hydrostatic Pressure
What is hydrostatic pressure?
Hydrostatic pressure is the pressure exerted by a column of fluid.
What generates the force that results in hydrostatic pressure?
Gravity.
How to calculate hydrostatic pressure?
It depends on the depth in the fluid the object is submerged and the shape of the object. But for any point in a water column, to calculate the pressure we use:
P=ρgh
Here h is the depth, g is the gravity and is the density ρ of the fluid.
What is the basic function of hydrostatic pressure?
P=ρgd
, where
P
is pressure,
ρ
is the density of the fluid,
g
is gravitational acceleration, and
d
or
h
is the depth below the surface.
What is the hydrostatic forces?
The hydrostatic force that acts on an area is
F=PA
, where
F
is force,
P
is hydrostatic pressure, and
A
is the area of the object.
How to measure hydrostatic pressure?
Usually with a gauge submerged in the fluid which returns the force per unit area.
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- Hydrostatic Pressure
# Hydrostatic Pressure
Have you ever wondered why divers need to be careful when going deep underwater, or when resurfacing? It is because the deeper you go, the higher the pressure the water exerts on the diver's body. To avoid the bends, divers stop at different depths to let the pressure of the water equalize with the pressure in their body. Read on for an introduction to the basics of hydrostatic pressure.
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[\+ Add tag Immunology Cell Biology Mo If you submerged two flat plates, one a square and one a circle, to the same depth, and the diameter of the circle is the same as the length of the side of the square, which would have lower hydrostatic pressure? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Which of the following would increase the hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Which of the following would increase the hydrostatic pressure? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Suppose you have a water column and you are measuring the hydrostatic force at three different heights: 1 meter, 2 meters, and 3 meters. Which measurement will give the lowest hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Suppose that both a professional football player and a small child went diving underwater to the same depth. Which one would experience greater hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Suppose you have a water column and you are measuring the hydrostatic force at three different heights: 1 meter, 2 meters, and 3 meters. Which measurement will give the greatest hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo True or False: Hydrostatic force does not depend on hydrostatic pressure. Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Hydrostatic \_\_\_\_ is a measurement involving a submerged area. Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Which of the techniques from calculus are used to help you find hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo When you go under water, you are experiencing \_\_\_\_. Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo If you submerged two flat plates, one a square and one a circle, to the same depth, and the diameter of the circle is the same as the length of the side of the square, which would have higher hydrostatic pressure? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Hydrostatic \_\_\_\_ is a measurement involving a submerged area. Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Which of the techniques from calculus are used to help you find hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo When you go under water, you are experiencing \_\_\_\_. Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo If you submerged two flat plates, one a square and one a circle, to the same depth, and the diameter of the circle is the same as the length of the side of the square, which would have higher hydrostatic pressure? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo If you submerged two flat plates, one a square and one a circle, to the same depth, and the diameter of the circle is the same as the length of the side of the square, which would have lower hydrostatic pressure? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Which of the following would increase the hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
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[\+ Add tag Immunology Cell Biology Mo Suppose you have a water column and you are measuring the hydrostatic force at three different heights: 1 meter, 2 meters, and 3 meters. Which measurement will give the lowest hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
[\+ Add tag Immunology Cell Biology Mo Suppose that both a professional football player and a small child went diving underwater to the same depth. Which one would experience greater hydrostatic force? Show Answer](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=hero-flashcard-section)
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- [Finding Limits of Specific Functions](https://www.vaia.com/en-us/explanations/math/calculus/finding-limits-of-specific-functions/)
- [First Derivative Test](https://www.vaia.com/en-us/explanations/math/calculus/first-derivative-test/)
- [Function Transformations](https://www.vaia.com/en-us/explanations/math/calculus/function-transformations/)
- [Fundamental Theorem of Line Integrals](https://www.vaia.com/en-us/explanations/math/calculus/fundamental-theorem-of-line-integrals/)
- [Galois Theory](https://www.vaia.com/en-us/explanations/math/calculus/galois-theory/)
- [General Solution of Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/general-solution-of-differential-equation/)
- [Geometric Series](https://www.vaia.com/en-us/explanations/math/calculus/geometric-series/)
- [Gradient Descent](https://www.vaia.com/en-us/explanations/math/calculus/gradient-descent/)
- [Graphing and Optimization](https://www.vaia.com/en-us/explanations/math/calculus/graphing-and-optimization/)
- [Green's Function](https://www.vaia.com/en-us/explanations/math/calculus/greens-function/)
- [Green's Theorem](https://www.vaia.com/en-us/explanations/math/calculus/greens-theorem/)
- [Growth Rate of Functions](https://www.vaia.com/en-us/explanations/math/calculus/growth-rate-of-functions/)
- [Harmonic Functions](https://www.vaia.com/en-us/explanations/math/calculus/harmonic-functions/)
- [Higher Order Partial Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/higher-order-partial-derivatives/)
- [Higher-Order Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/higher-order-derivatives/)
- [Hydrostatic Pressure](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/)
- [Hyperbolic Functions](https://www.vaia.com/en-us/explanations/math/calculus/hyperbolic-functions/)
- [Implicit Differentiation Tangent Line](https://www.vaia.com/en-us/explanations/math/calculus/implicit-differentiation-tangent-line/)
- [Implicit Relations](https://www.vaia.com/en-us/explanations/math/calculus/implicit-relations/)
- [Improper Integrals](https://www.vaia.com/en-us/explanations/math/calculus/improper-integrals/)
- [Increasing and Decreasing Functions](https://www.vaia.com/en-us/explanations/math/calculus/increasing-and-decreasing-functions/)
- [Indefinite Integral](https://www.vaia.com/en-us/explanations/math/calculus/indefinite-integral/)
- [Indeterminate Forms](https://www.vaia.com/en-us/explanations/math/calculus/indeterminate-forms/)
- [Indeterminate Forms of Limits](https://www.vaia.com/en-us/explanations/math/calculus/indeterminate-forms-of-limits/)
- [Initial Value Problem Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/initial-value-problem-differential-equations/)
- [Integral Equations](https://www.vaia.com/en-us/explanations/math/calculus/integral-equations/)
- [Integral Test](https://www.vaia.com/en-us/explanations/math/calculus/integral-test/)
- [Integrals in Economics](https://www.vaia.com/en-us/explanations/math/calculus/integrals-in-economics/)
- [Integrals of Exponential Functions](https://www.vaia.com/en-us/explanations/math/calculus/integrals-of-exponential-functions/)
- [Integrals of Motion](https://www.vaia.com/en-us/explanations/math/calculus/integrals-of-motion/)
- [Integrating Even and Odd Functions](https://www.vaia.com/en-us/explanations/math/calculus/integrating-even-and-odd-functions/)
- [Integration Formula](https://www.vaia.com/en-us/explanations/math/calculus/integration-formula/)
- [Integration Tables](https://www.vaia.com/en-us/explanations/math/calculus/integration-tables/)
- [Integration Techniques](https://www.vaia.com/en-us/explanations/math/calculus/integration-techniques/)
- [Integration Using Long Division](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-long-division/)
- [Integration Using Tables](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-tables/)
- [Integration fundamentals](https://www.vaia.com/en-us/explanations/math/calculus/integration-fundamentals/)
- [Integration of Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-of-logarithmic-functions/)
- [Integration of Vector Valued Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-of-vector-valued-functions/)
- [Integration using Inverse Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-inverse-trigonometric-functions/)
- [Intermediate Value Theorem](https://www.vaia.com/en-us/explanations/math/calculus/intermediate-value-theorem/)
- [Inverse Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/inverse-trigonometric-functions/)
- [Jump Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/jump-discontinuity/)
- [Lagrange Error Bound](https://www.vaia.com/en-us/explanations/math/calculus/lagrange-error-bound/)
- [Lagrange Multiplier](https://www.vaia.com/en-us/explanations/math/calculus/lagrange-multiplier/)
- [Laplace Transform](https://www.vaia.com/en-us/explanations/math/calculus/laplace-transform/)
- [Lebesgue Integration](https://www.vaia.com/en-us/explanations/math/calculus/lebesgue-integration/)
- [Limit Applications](https://www.vaia.com/en-us/explanations/math/calculus/limit-applications/)
- [Limit Laws](https://www.vaia.com/en-us/explanations/math/calculus/limit-laws/)
- [Limit of Vector Valued Function](https://www.vaia.com/en-us/explanations/math/calculus/limit-of-vector-valued-function/)
- [Limit of a Sequence](https://www.vaia.com/en-us/explanations/math/calculus/limit-of-a-sequence/)
- [Limits](https://www.vaia.com/en-us/explanations/math/calculus/limits/)
- [Limits and Continuity](https://www.vaia.com/en-us/explanations/math/calculus/limits-and-continuity/)
- [Limits at Infinity](https://www.vaia.com/en-us/explanations/math/calculus/limits-at-infinity/)
- [Limits at Infinity and Asymptotes](https://www.vaia.com/en-us/explanations/math/calculus/limits-at-infinity-and-asymptotes/)
- [Limits of a Function](https://www.vaia.com/en-us/explanations/math/calculus/limits-of-a-function/)
- [Line Integrals](https://www.vaia.com/en-us/explanations/math/calculus/line-integrals/)
- [Linear Approximations and Differentials](https://www.vaia.com/en-us/explanations/math/calculus/linear-approximations-and-differentials/)
- [Linear Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/linear-differential-equation/)
- [Linear Functions](https://www.vaia.com/en-us/explanations/math/calculus/linear-functions/)
- [Logarithmic Differentiation](https://www.vaia.com/en-us/explanations/math/calculus/logarithmic-differentiation/)
- [Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/logarithmic-functions/)
- [Logistic Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/logistic-differential-equation/)
- [Logistic Model](https://www.vaia.com/en-us/explanations/math/calculus/logistic-model/)
- [Maclaurin Series](https://www.vaia.com/en-us/explanations/math/calculus/maclaurin-series/)
- [Manipulating Functions](https://www.vaia.com/en-us/explanations/math/calculus/manipulating-functions/)
- [Matrix Theory](https://www.vaia.com/en-us/explanations/math/calculus/matrix-theory/)
- [Maxima and Minima](https://www.vaia.com/en-us/explanations/math/calculus/maxima-and-minima/)
- [Maxima and Minima Problems](https://www.vaia.com/en-us/explanations/math/calculus/maxima-and-minima-problems/)
- [Mean Value Theorem for Integrals](https://www.vaia.com/en-us/explanations/math/calculus/mean-value-theorem-for-integrals/)
- [Measure Theory](https://www.vaia.com/en-us/explanations/math/calculus/measure-theory/)
- [Michaelis Menten Equation](https://www.vaia.com/en-us/explanations/math/calculus/michaelis-menten-equation/)
- [Models for Population Growth](https://www.vaia.com/en-us/explanations/math/calculus/models-for-population-growth/)
- [Motion Along A Line](https://www.vaia.com/en-us/explanations/math/calculus/motion-along-a-line/)
- [Motion In Space](https://www.vaia.com/en-us/explanations/math/calculus/motion-in-space/)
- [Multiple Integrals](https://www.vaia.com/en-us/explanations/math/calculus/multiple-integrals/)
- [Multivariable Calculus](https://www.vaia.com/en-us/explanations/math/calculus/multivariable-calculus/)
- [Natural Logarithmic Function](https://www.vaia.com/en-us/explanations/math/calculus/natural-logarithmic-function/)
- [Net Change Theorem](https://www.vaia.com/en-us/explanations/math/calculus/net-change-theorem/)
- [Newton's Method](https://www.vaia.com/en-us/explanations/math/calculus/newtons-method/)
- [Non Differentiable Functions](https://www.vaia.com/en-us/explanations/math/calculus/non-differentiable-functions/)
- [Nonhomogeneous Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/nonhomogeneous-differential-equation/)
- [Nonlinear Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/nonlinear-differential-equations/)
- [Numerical Linear Algebra](https://www.vaia.com/en-us/explanations/math/calculus/numerical-linear-algebra/)
- [One-Sided Limits](https://www.vaia.com/en-us/explanations/math/calculus/one-sided-limits/)
- [Optimization Problems](https://www.vaia.com/en-us/explanations/math/calculus/optimization-problems/)
- [Optimization Problems in Economics](https://www.vaia.com/en-us/explanations/math/calculus/optimization-problems-in-economics/)
- [Optimization Theory](https://www.vaia.com/en-us/explanations/math/calculus/optimization-theory/)
- [Ordinary Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/ordinary-differential-equations/)
- [P Series](https://www.vaia.com/en-us/explanations/math/calculus/p-series/)
- [PDE Solutions](https://www.vaia.com/en-us/explanations/math/calculus/pde-solutions/)
- [Parametric Surface Area](https://www.vaia.com/en-us/explanations/math/calculus/parametric-surface-area/)
- [Parametric derivatives](https://www.vaia.com/en-us/explanations/math/calculus/parametric-derivatives/)
- [Partial Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/partial-derivatives/)
- [Particle Model Motion](https://www.vaia.com/en-us/explanations/math/calculus/particle-model-motion/)
- [Particular Solutions to Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/particular-solutions-to-differential-equations/)
- [Piecewise Defined Function](https://www.vaia.com/en-us/explanations/math/calculus/piecewise-defined-function/)
- [Polar Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/polar-coordinates/)
- [Polar Coordinates Functions](https://www.vaia.com/en-us/explanations/math/calculus/polar-coordinates-functions/)
- [Polar Curves](https://www.vaia.com/en-us/explanations/math/calculus/polar-curves/)
- [Population Change](https://www.vaia.com/en-us/explanations/math/calculus/population-change/)
- [Power Series](https://www.vaia.com/en-us/explanations/math/calculus/power-series/)
- [Probability Theory](https://www.vaia.com/en-us/explanations/math/calculus/probability-theory/)
- [Properties of Definite Integrals](https://www.vaia.com/en-us/explanations/math/calculus/properties-of-definite-integrals/)
- [Radius of Convergence](https://www.vaia.com/en-us/explanations/math/calculus/radius-of-convergence/)
- [Ratio Test](https://www.vaia.com/en-us/explanations/math/calculus/ratio-test/)
- [Real Analysis](https://www.vaia.com/en-us/explanations/math/calculus/real-analysis/)
- [Related Rates](https://www.vaia.com/en-us/explanations/math/calculus/related-rates/)
- [Removable Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/removable-discontinuity/)
- [Revenue as Average Rate of Change](https://www.vaia.com/en-us/explanations/math/calculus/revenue-as-average-rate-of-change/)
- [Riemann Integral](https://www.vaia.com/en-us/explanations/math/calculus/riemann-integral/)
- [Riemann Sum](https://www.vaia.com/en-us/explanations/math/calculus/riemann-sum/)
- [Rolle's Theorem](https://www.vaia.com/en-us/explanations/math/calculus/rolles-theorem/)
- [Root Test](https://www.vaia.com/en-us/explanations/math/calculus/root-test/)
- [Second Derivative Test](https://www.vaia.com/en-us/explanations/math/calculus/second-derivative-test/)
- [Separable Equations](https://www.vaia.com/en-us/explanations/math/calculus/separable-equations/)
- [Separable differential equations](https://www.vaia.com/en-us/explanations/math/calculus/separable-differential-equations/)
- [Separation of Variables](https://www.vaia.com/en-us/explanations/math/calculus/separation-of-variables/)
- [Simpson's Rule](https://www.vaia.com/en-us/explanations/math/calculus/simpsons-rule/)
- [Slope Fields](https://www.vaia.com/en-us/explanations/math/calculus/slope-fields/)
- [Solid of Revolution](https://www.vaia.com/en-us/explanations/math/calculus/solid-of-revolution/)
- [Solutions to Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/solutions-to-differential-equations/)
- [Solving Inequalities using Continuity Properties](https://www.vaia.com/en-us/explanations/math/calculus/solving-inequalities-using-continuity-properties/)
- [Spectral Theory](https://www.vaia.com/en-us/explanations/math/calculus/spectral-theory/)
- [Stochastic Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/stochastic-differential-equations/)
- [Surface Area Integral](https://www.vaia.com/en-us/explanations/math/calculus/surface-area-integral/)
- [Surface Area of Revolution](https://www.vaia.com/en-us/explanations/math/calculus/surface-area-of-revolution/)
- [Surface Integrals](https://www.vaia.com/en-us/explanations/math/calculus/surface-integrals/)
- [Surplus](https://www.vaia.com/en-us/explanations/math/calculus/surplus/)
- [Symmetry of Functions](https://www.vaia.com/en-us/explanations/math/calculus/symmetry-of-functions/)
- [Tangent Lines](https://www.vaia.com/en-us/explanations/math/calculus/tangent-lines/)
- [Tangent Plane](https://www.vaia.com/en-us/explanations/math/calculus/tangent-plane/)
- [Tangent Planes and Linear Approximations](https://www.vaia.com/en-us/explanations/math/calculus/tangent-planes-and-linear-approximations/)
- [Taylor Polynomials](https://www.vaia.com/en-us/explanations/math/calculus/taylor-polynomials/)
- [Taylor Series](https://www.vaia.com/en-us/explanations/math/calculus/taylor-series/)
- [Techniques of Integration](https://www.vaia.com/en-us/explanations/math/calculus/techniques-of-integration/)
- [The Fundamental Theorem of Calculus](https://www.vaia.com/en-us/explanations/math/calculus/the-fundamental-theorem-of-calculus/)
- [The Limit Does Not Exist](https://www.vaia.com/en-us/explanations/math/calculus/the-limit-does-not-exist/)
- [The Mean Value Theorem](https://www.vaia.com/en-us/explanations/math/calculus/the-mean-value-theorem/)
- [The Power Rule](https://www.vaia.com/en-us/explanations/math/calculus/the-power-rule/)
- [The Squeeze Theorem](https://www.vaia.com/en-us/explanations/math/calculus/the-squeeze-theorem/)
- [The Trapezoidal Rule](https://www.vaia.com/en-us/explanations/math/calculus/the-trapezoidal-rule/)
- [Theorems of Continuity](https://www.vaia.com/en-us/explanations/math/calculus/theorems-of-continuity/)
- [Topology](https://www.vaia.com/en-us/explanations/math/calculus/topology/)
- [Trigonometric Substitution](https://www.vaia.com/en-us/explanations/math/calculus/trigonometric-substitution/)
- [Triple Integral](https://www.vaia.com/en-us/explanations/math/calculus/triple-integral/)
- [Triple Integral Spherical Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/triple-integral-spherical-coordinates/)
- [Triple Integrals in Cylindrical Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/triple-integrals-in-cylindrical-coordinates/)
- [Types of Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/types-of-discontinuity/)
- [Using Slope Fields to Graph Solutions](https://www.vaia.com/en-us/explanations/math/calculus/using-slope-fields-to-graph-solutions/)
- [Variational Methods](https://www.vaia.com/en-us/explanations/math/calculus/variational-methods/)
- [Vector Valued Function](https://www.vaia.com/en-us/explanations/math/calculus/vector-valued-function/)
- [Vectors In Space](https://www.vaia.com/en-us/explanations/math/calculus/vectors-in-space/)
- [Vectors in Calculus](https://www.vaia.com/en-us/explanations/math/calculus/vectors-in-calculus/)
- [Velocity as Average Rate of Change](https://www.vaia.com/en-us/explanations/math/calculus/velocity-as-average-rate-of-change/)
- [Vertical Asymptote](https://www.vaia.com/en-us/explanations/math/calculus/vertical-asymptote/)
- [Volume Integrals](https://www.vaia.com/en-us/explanations/math/calculus/volume-integrals/)
- [Volume by disks](https://www.vaia.com/en-us/explanations/math/calculus/volume-by-disks/)
- [Volume by shells](https://www.vaia.com/en-us/explanations/math/calculus/volume-by-shells/)
- [Washer Method](https://www.vaia.com/en-us/explanations/math/calculus/washer-method/)
- [Decision Maths](https://www.vaia.com/en-us/explanations/math/decision-maths/)
- [Discrete Mathematics](https://www.vaia.com/en-us/explanations/math/discrete-mathematics/)
- [Geometry](https://www.vaia.com/en-us/explanations/math/geometry/)
- [Logic and Functions](https://www.vaia.com/en-us/explanations/math/logic-and-functions/)
- [Mechanics Maths](https://www.vaia.com/en-us/explanations/math/mechanics-maths/)
- [Probability and Statistics](https://www.vaia.com/en-us/explanations/math/probability-and-statistics/)
- [Pure Maths](https://www.vaia.com/en-us/explanations/math/pure-maths/)
- [Statistics](https://www.vaia.com/en-us/explanations/math/statistics/)
- [Theoretical and Mathematical Physics](https://www.vaia.com/en-us/explanations/math/theoretical-and-mathematical-physics/)
Contents
- [Applied Mathematics](https://www.vaia.com/en-us/explanations/math/applied-mathematics/)
- [Calculus](https://www.vaia.com/en-us/explanations/math/calculus/)
- [Absolute Maxima and Minima](https://www.vaia.com/en-us/explanations/math/calculus/absolute-maxima-and-minima/)
- [Absolute and Conditional Convergence](https://www.vaia.com/en-us/explanations/math/calculus/absolute-and-conditional-convergence/)
- [Accumulation Function](https://www.vaia.com/en-us/explanations/math/calculus/accumulation-function/)
- [Accumulation Problems](https://www.vaia.com/en-us/explanations/math/calculus/accumulation-problems/)
- [Algebraic Functions](https://www.vaia.com/en-us/explanations/math/calculus/algebraic-functions/)
- [Alternating Series](https://www.vaia.com/en-us/explanations/math/calculus/alternating-series/)
- [Antiderivatives](https://www.vaia.com/en-us/explanations/math/calculus/antiderivatives/)
- [Application of Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/application-of-derivatives/)
- [Application of Higher Order Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/application-of-higher-order-derivatives/)
- [Application of Integrals in Biology and Social Sciences](https://www.vaia.com/en-us/explanations/math/calculus/application-of-integrals-in-biology-and-social-sciences/)
- [Applications of Continuity](https://www.vaia.com/en-us/explanations/math/calculus/applications-of-continuity/)
- [Applications of Double Integrals](https://www.vaia.com/en-us/explanations/math/calculus/applications-of-double-integrals/)
- [Approximating Areas](https://www.vaia.com/en-us/explanations/math/calculus/approximating-areas/)
- [Arc Length of a Curve](https://www.vaia.com/en-us/explanations/math/calculus/arc-length-of-a-curve/)
- [Area Between Two Curves](https://www.vaia.com/en-us/explanations/math/calculus/area-between-two-curves/)
- [Arithmetic Series](https://www.vaia.com/en-us/explanations/math/calculus/arithmetic-series/)
- [Arithmetic of Complex Numbers](https://www.vaia.com/en-us/explanations/math/calculus/arithmetic-of-complex-numbers/)
- [Average Rate of Change of Populations](https://www.vaia.com/en-us/explanations/math/calculus/average-rate-of-change-of-populations/)
- [Average Value Function](https://www.vaia.com/en-us/explanations/math/calculus/average-value-function/)
- [Average Value of a Function](https://www.vaia.com/en-us/explanations/math/calculus/average-value-of-a-function/)
- [Bifurcation Theory](https://www.vaia.com/en-us/explanations/math/calculus/bifurcation-theory/)
- [Boundary Value Problems](https://www.vaia.com/en-us/explanations/math/calculus/boundary-value-problems/)
- [Calculus Linear Approximation](https://www.vaia.com/en-us/explanations/math/calculus/calculus-linear-approximation/)
- [Calculus Of Parametric Curves](https://www.vaia.com/en-us/explanations/math/calculus/calculus-of-parametric-curves/)
- [Candidate Test](https://www.vaia.com/en-us/explanations/math/calculus/candidate-test/)
- [Change of Variables in Multiple Integrals](https://www.vaia.com/en-us/explanations/math/calculus/change-of-variables-in-multiple-integrals/)
- [Combining Different Rules](https://www.vaia.com/en-us/explanations/math/calculus/combining-different-rules/)
- [Combining Differentiation Rules](https://www.vaia.com/en-us/explanations/math/calculus/combining-differentiation-rules/)
- [Combining Functions](https://www.vaia.com/en-us/explanations/math/calculus/combining-functions/)
- [Complex Analysis](https://www.vaia.com/en-us/explanations/math/calculus/complex-analysis/)
- [Concavity of a Function](https://www.vaia.com/en-us/explanations/math/calculus/concavity-of-a-function/)
- [Continuity](https://www.vaia.com/en-us/explanations/math/calculus/continuity/)
- [Continuity Equations](https://www.vaia.com/en-us/explanations/math/calculus/continuity-equations/)
- [Continuity Over an Interval](https://www.vaia.com/en-us/explanations/math/calculus/continuity-over-an-interval/)
- [Continuity and Indeterminate Forms](https://www.vaia.com/en-us/explanations/math/calculus/continuity-and-indeterminate-forms/)
- [Convergence Tests](https://www.vaia.com/en-us/explanations/math/calculus/convergence-tests/)
- [Cost and Revenue](https://www.vaia.com/en-us/explanations/math/calculus/cost-and-revenue/)
- [Critical points](https://www.vaia.com/en-us/explanations/math/calculus/critical-points/)
- [Curl and Divergence](https://www.vaia.com/en-us/explanations/math/calculus/curl-and-divergence/)
- [Curve Sketching Techniques](https://www.vaia.com/en-us/explanations/math/calculus/curve-sketching-techniques/)
- [Density and Center of Mass](https://www.vaia.com/en-us/explanations/math/calculus/density-and-center-of-mass/)
- [Derivative Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivative-functions/)
- [Derivative Properties](https://www.vaia.com/en-us/explanations/math/calculus/derivative-properties/)
- [Derivative as a Limit](https://www.vaia.com/en-us/explanations/math/calculus/derivative-as-a-limit/)
- [Derivative of Exponential Function](https://www.vaia.com/en-us/explanations/math/calculus/derivative-of-exponential-function/)
- [Derivative of Inverse Function](https://www.vaia.com/en-us/explanations/math/calculus/derivative-of-inverse-function/)
- [Derivative of Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivative-of-logarithmic-functions/)
- [Derivative of Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivative-of-trigonometric-functions/)
- [Derivative of Vector Valued Function](https://www.vaia.com/en-us/explanations/math/calculus/derivative-of-vector-valued-function/)
- [Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/derivatives/)
- [Derivatives and Continuity](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-and-continuity/)
- [Derivatives and the Shape of a Graph](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-and-the-shape-of-a-graph/)
- [Derivatives of Exponential Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-exponential-functions/)
- [Derivatives of Inverse Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-inverse-trigonometric-functions/)
- [Derivatives of Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-logarithmic-functions/)
- [Derivatives of Polar Functions](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-polar-functions/)
- [Derivatives of Sec, Csc and Cot](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-sec-csc-and-cot/)
- [Derivatives of Sin, Cos and Tan](https://www.vaia.com/en-us/explanations/math/calculus/derivatives-of-sin-cos-and-tan/)
- [Determining Volumes by Slicing](https://www.vaia.com/en-us/explanations/math/calculus/determining-volumes-by-slicing/)
- [Difference Quotient](https://www.vaia.com/en-us/explanations/math/calculus/difference-quotient/)
- [Differentiability](https://www.vaia.com/en-us/explanations/math/calculus/differentiability/)
- [Differentials](https://www.vaia.com/en-us/explanations/math/calculus/differentials/)
- [Differentiation of Functions of Several Variables](https://www.vaia.com/en-us/explanations/math/calculus/differentiation-of-functions-of-several-variables/)
- [Direction Fields](https://www.vaia.com/en-us/explanations/math/calculus/direction-fields/)
- [Disk Method](https://www.vaia.com/en-us/explanations/math/calculus/the-disk-method/)
- [Divergence Test](https://www.vaia.com/en-us/explanations/math/calculus/divergence-test/)
- [Double Integral](https://www.vaia.com/en-us/explanations/math/calculus/double-integral/)
- [Double Integrals Over General Regions](https://www.vaia.com/en-us/explanations/math/calculus/double-integrals-over-general-regions/)
- [Double Integrals Over Rectangular Regions](https://www.vaia.com/en-us/explanations/math/calculus/double-integrals-over-rectangular-regions/)
- [Double Integrals in Polar Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/double-integrals-in-polar-coordinates/)
- [Dynamical Systems](https://www.vaia.com/en-us/explanations/math/calculus/dynamical-systems/)
- [Eliminating The Parameter](https://www.vaia.com/en-us/explanations/math/calculus/eliminating-the-parameter/)
- [Euler's Method](https://www.vaia.com/en-us/explanations/math/calculus/eulers-method/)
- [Evaluating a Definite Integral](https://www.vaia.com/en-us/explanations/math/calculus/evaluating-a-definite-integral/)
- [Evaluation Theorem](https://www.vaia.com/en-us/explanations/math/calculus/evaluation-theorem/)
- [Exact equations](https://www.vaia.com/en-us/explanations/math/calculus/exact-equations/)
- [Exponential Functions](https://www.vaia.com/en-us/explanations/math/calculus/exponential-functions/)
- [Exponential Model](https://www.vaia.com/en-us/explanations/math/calculus/exponential-model/)
- [Extrema](https://www.vaia.com/en-us/explanations/math/calculus/extrema/)
- [Finding Limits](https://www.vaia.com/en-us/explanations/math/calculus/finding-limits/)
- [Finding Limits of Specific Functions](https://www.vaia.com/en-us/explanations/math/calculus/finding-limits-of-specific-functions/)
- [First Derivative Test](https://www.vaia.com/en-us/explanations/math/calculus/first-derivative-test/)
- [Function Transformations](https://www.vaia.com/en-us/explanations/math/calculus/function-transformations/)
- [Fundamental Theorem of Line Integrals](https://www.vaia.com/en-us/explanations/math/calculus/fundamental-theorem-of-line-integrals/)
- [Galois Theory](https://www.vaia.com/en-us/explanations/math/calculus/galois-theory/)
- [General Solution of Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/general-solution-of-differential-equation/)
- [Geometric Series](https://www.vaia.com/en-us/explanations/math/calculus/geometric-series/)
- [Gradient Descent](https://www.vaia.com/en-us/explanations/math/calculus/gradient-descent/)
- [Graphing and Optimization](https://www.vaia.com/en-us/explanations/math/calculus/graphing-and-optimization/)
- [Green's Function](https://www.vaia.com/en-us/explanations/math/calculus/greens-function/)
- [Green's Theorem](https://www.vaia.com/en-us/explanations/math/calculus/greens-theorem/)
- [Growth Rate of Functions](https://www.vaia.com/en-us/explanations/math/calculus/growth-rate-of-functions/)
- [Harmonic Functions](https://www.vaia.com/en-us/explanations/math/calculus/harmonic-functions/)
- [Higher Order Partial Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/higher-order-partial-derivatives/)
- [Higher-Order Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/higher-order-derivatives/)
- [Hydrostatic Pressure](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/)
- [Hyperbolic Functions](https://www.vaia.com/en-us/explanations/math/calculus/hyperbolic-functions/)
- [Implicit Differentiation Tangent Line](https://www.vaia.com/en-us/explanations/math/calculus/implicit-differentiation-tangent-line/)
- [Implicit Relations](https://www.vaia.com/en-us/explanations/math/calculus/implicit-relations/)
- [Improper Integrals](https://www.vaia.com/en-us/explanations/math/calculus/improper-integrals/)
- [Increasing and Decreasing Functions](https://www.vaia.com/en-us/explanations/math/calculus/increasing-and-decreasing-functions/)
- [Indefinite Integral](https://www.vaia.com/en-us/explanations/math/calculus/indefinite-integral/)
- [Indeterminate Forms](https://www.vaia.com/en-us/explanations/math/calculus/indeterminate-forms/)
- [Indeterminate Forms of Limits](https://www.vaia.com/en-us/explanations/math/calculus/indeterminate-forms-of-limits/)
- [Initial Value Problem Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/initial-value-problem-differential-equations/)
- [Integral Equations](https://www.vaia.com/en-us/explanations/math/calculus/integral-equations/)
- [Integral Test](https://www.vaia.com/en-us/explanations/math/calculus/integral-test/)
- [Integrals in Economics](https://www.vaia.com/en-us/explanations/math/calculus/integrals-in-economics/)
- [Integrals of Exponential Functions](https://www.vaia.com/en-us/explanations/math/calculus/integrals-of-exponential-functions/)
- [Integrals of Motion](https://www.vaia.com/en-us/explanations/math/calculus/integrals-of-motion/)
- [Integrating Even and Odd Functions](https://www.vaia.com/en-us/explanations/math/calculus/integrating-even-and-odd-functions/)
- [Integration Formula](https://www.vaia.com/en-us/explanations/math/calculus/integration-formula/)
- [Integration Tables](https://www.vaia.com/en-us/explanations/math/calculus/integration-tables/)
- [Integration Techniques](https://www.vaia.com/en-us/explanations/math/calculus/integration-techniques/)
- [Integration Using Long Division](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-long-division/)
- [Integration Using Tables](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-tables/)
- [Integration fundamentals](https://www.vaia.com/en-us/explanations/math/calculus/integration-fundamentals/)
- [Integration of Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-of-logarithmic-functions/)
- [Integration of Vector Valued Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-of-vector-valued-functions/)
- [Integration using Inverse Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/integration-using-inverse-trigonometric-functions/)
- [Intermediate Value Theorem](https://www.vaia.com/en-us/explanations/math/calculus/intermediate-value-theorem/)
- [Inverse Trigonometric Functions](https://www.vaia.com/en-us/explanations/math/calculus/inverse-trigonometric-functions/)
- [Jump Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/jump-discontinuity/)
- [Lagrange Error Bound](https://www.vaia.com/en-us/explanations/math/calculus/lagrange-error-bound/)
- [Lagrange Multiplier](https://www.vaia.com/en-us/explanations/math/calculus/lagrange-multiplier/)
- [Laplace Transform](https://www.vaia.com/en-us/explanations/math/calculus/laplace-transform/)
- [Lebesgue Integration](https://www.vaia.com/en-us/explanations/math/calculus/lebesgue-integration/)
- [Limit Applications](https://www.vaia.com/en-us/explanations/math/calculus/limit-applications/)
- [Limit Laws](https://www.vaia.com/en-us/explanations/math/calculus/limit-laws/)
- [Limit of Vector Valued Function](https://www.vaia.com/en-us/explanations/math/calculus/limit-of-vector-valued-function/)
- [Limit of a Sequence](https://www.vaia.com/en-us/explanations/math/calculus/limit-of-a-sequence/)
- [Limits](https://www.vaia.com/en-us/explanations/math/calculus/limits/)
- [Limits and Continuity](https://www.vaia.com/en-us/explanations/math/calculus/limits-and-continuity/)
- [Limits at Infinity](https://www.vaia.com/en-us/explanations/math/calculus/limits-at-infinity/)
- [Limits at Infinity and Asymptotes](https://www.vaia.com/en-us/explanations/math/calculus/limits-at-infinity-and-asymptotes/)
- [Limits of a Function](https://www.vaia.com/en-us/explanations/math/calculus/limits-of-a-function/)
- [Line Integrals](https://www.vaia.com/en-us/explanations/math/calculus/line-integrals/)
- [Linear Approximations and Differentials](https://www.vaia.com/en-us/explanations/math/calculus/linear-approximations-and-differentials/)
- [Linear Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/linear-differential-equation/)
- [Linear Functions](https://www.vaia.com/en-us/explanations/math/calculus/linear-functions/)
- [Logarithmic Differentiation](https://www.vaia.com/en-us/explanations/math/calculus/logarithmic-differentiation/)
- [Logarithmic Functions](https://www.vaia.com/en-us/explanations/math/calculus/logarithmic-functions/)
- [Logistic Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/logistic-differential-equation/)
- [Logistic Model](https://www.vaia.com/en-us/explanations/math/calculus/logistic-model/)
- [Maclaurin Series](https://www.vaia.com/en-us/explanations/math/calculus/maclaurin-series/)
- [Manipulating Functions](https://www.vaia.com/en-us/explanations/math/calculus/manipulating-functions/)
- [Matrix Theory](https://www.vaia.com/en-us/explanations/math/calculus/matrix-theory/)
- [Maxima and Minima](https://www.vaia.com/en-us/explanations/math/calculus/maxima-and-minima/)
- [Maxima and Minima Problems](https://www.vaia.com/en-us/explanations/math/calculus/maxima-and-minima-problems/)
- [Mean Value Theorem for Integrals](https://www.vaia.com/en-us/explanations/math/calculus/mean-value-theorem-for-integrals/)
- [Measure Theory](https://www.vaia.com/en-us/explanations/math/calculus/measure-theory/)
- [Michaelis Menten Equation](https://www.vaia.com/en-us/explanations/math/calculus/michaelis-menten-equation/)
- [Models for Population Growth](https://www.vaia.com/en-us/explanations/math/calculus/models-for-population-growth/)
- [Motion Along A Line](https://www.vaia.com/en-us/explanations/math/calculus/motion-along-a-line/)
- [Motion In Space](https://www.vaia.com/en-us/explanations/math/calculus/motion-in-space/)
- [Multiple Integrals](https://www.vaia.com/en-us/explanations/math/calculus/multiple-integrals/)
- [Multivariable Calculus](https://www.vaia.com/en-us/explanations/math/calculus/multivariable-calculus/)
- [Natural Logarithmic Function](https://www.vaia.com/en-us/explanations/math/calculus/natural-logarithmic-function/)
- [Net Change Theorem](https://www.vaia.com/en-us/explanations/math/calculus/net-change-theorem/)
- [Newton's Method](https://www.vaia.com/en-us/explanations/math/calculus/newtons-method/)
- [Non Differentiable Functions](https://www.vaia.com/en-us/explanations/math/calculus/non-differentiable-functions/)
- [Nonhomogeneous Differential Equation](https://www.vaia.com/en-us/explanations/math/calculus/nonhomogeneous-differential-equation/)
- [Nonlinear Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/nonlinear-differential-equations/)
- [Numerical Linear Algebra](https://www.vaia.com/en-us/explanations/math/calculus/numerical-linear-algebra/)
- [One-Sided Limits](https://www.vaia.com/en-us/explanations/math/calculus/one-sided-limits/)
- [Optimization Problems](https://www.vaia.com/en-us/explanations/math/calculus/optimization-problems/)
- [Optimization Problems in Economics](https://www.vaia.com/en-us/explanations/math/calculus/optimization-problems-in-economics/)
- [Optimization Theory](https://www.vaia.com/en-us/explanations/math/calculus/optimization-theory/)
- [Ordinary Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/ordinary-differential-equations/)
- [P Series](https://www.vaia.com/en-us/explanations/math/calculus/p-series/)
- [PDE Solutions](https://www.vaia.com/en-us/explanations/math/calculus/pde-solutions/)
- [Parametric Surface Area](https://www.vaia.com/en-us/explanations/math/calculus/parametric-surface-area/)
- [Parametric derivatives](https://www.vaia.com/en-us/explanations/math/calculus/parametric-derivatives/)
- [Partial Derivatives](https://www.vaia.com/en-us/explanations/math/calculus/partial-derivatives/)
- [Particle Model Motion](https://www.vaia.com/en-us/explanations/math/calculus/particle-model-motion/)
- [Particular Solutions to Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/particular-solutions-to-differential-equations/)
- [Piecewise Defined Function](https://www.vaia.com/en-us/explanations/math/calculus/piecewise-defined-function/)
- [Polar Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/polar-coordinates/)
- [Polar Coordinates Functions](https://www.vaia.com/en-us/explanations/math/calculus/polar-coordinates-functions/)
- [Polar Curves](https://www.vaia.com/en-us/explanations/math/calculus/polar-curves/)
- [Population Change](https://www.vaia.com/en-us/explanations/math/calculus/population-change/)
- [Power Series](https://www.vaia.com/en-us/explanations/math/calculus/power-series/)
- [Probability Theory](https://www.vaia.com/en-us/explanations/math/calculus/probability-theory/)
- [Properties of Definite Integrals](https://www.vaia.com/en-us/explanations/math/calculus/properties-of-definite-integrals/)
- [Radius of Convergence](https://www.vaia.com/en-us/explanations/math/calculus/radius-of-convergence/)
- [Ratio Test](https://www.vaia.com/en-us/explanations/math/calculus/ratio-test/)
- [Real Analysis](https://www.vaia.com/en-us/explanations/math/calculus/real-analysis/)
- [Related Rates](https://www.vaia.com/en-us/explanations/math/calculus/related-rates/)
- [Removable Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/removable-discontinuity/)
- [Revenue as Average Rate of Change](https://www.vaia.com/en-us/explanations/math/calculus/revenue-as-average-rate-of-change/)
- [Riemann Integral](https://www.vaia.com/en-us/explanations/math/calculus/riemann-integral/)
- [Riemann Sum](https://www.vaia.com/en-us/explanations/math/calculus/riemann-sum/)
- [Rolle's Theorem](https://www.vaia.com/en-us/explanations/math/calculus/rolles-theorem/)
- [Root Test](https://www.vaia.com/en-us/explanations/math/calculus/root-test/)
- [Second Derivative Test](https://www.vaia.com/en-us/explanations/math/calculus/second-derivative-test/)
- [Separable Equations](https://www.vaia.com/en-us/explanations/math/calculus/separable-equations/)
- [Separable differential equations](https://www.vaia.com/en-us/explanations/math/calculus/separable-differential-equations/)
- [Separation of Variables](https://www.vaia.com/en-us/explanations/math/calculus/separation-of-variables/)
- [Simpson's Rule](https://www.vaia.com/en-us/explanations/math/calculus/simpsons-rule/)
- [Slope Fields](https://www.vaia.com/en-us/explanations/math/calculus/slope-fields/)
- [Solid of Revolution](https://www.vaia.com/en-us/explanations/math/calculus/solid-of-revolution/)
- [Solutions to Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/solutions-to-differential-equations/)
- [Solving Inequalities using Continuity Properties](https://www.vaia.com/en-us/explanations/math/calculus/solving-inequalities-using-continuity-properties/)
- [Spectral Theory](https://www.vaia.com/en-us/explanations/math/calculus/spectral-theory/)
- [Stochastic Differential Equations](https://www.vaia.com/en-us/explanations/math/calculus/stochastic-differential-equations/)
- [Surface Area Integral](https://www.vaia.com/en-us/explanations/math/calculus/surface-area-integral/)
- [Surface Area of Revolution](https://www.vaia.com/en-us/explanations/math/calculus/surface-area-of-revolution/)
- [Surface Integrals](https://www.vaia.com/en-us/explanations/math/calculus/surface-integrals/)
- [Surplus](https://www.vaia.com/en-us/explanations/math/calculus/surplus/)
- [Symmetry of Functions](https://www.vaia.com/en-us/explanations/math/calculus/symmetry-of-functions/)
- [Tangent Lines](https://www.vaia.com/en-us/explanations/math/calculus/tangent-lines/)
- [Tangent Plane](https://www.vaia.com/en-us/explanations/math/calculus/tangent-plane/)
- [Tangent Planes and Linear Approximations](https://www.vaia.com/en-us/explanations/math/calculus/tangent-planes-and-linear-approximations/)
- [Taylor Polynomials](https://www.vaia.com/en-us/explanations/math/calculus/taylor-polynomials/)
- [Taylor Series](https://www.vaia.com/en-us/explanations/math/calculus/taylor-series/)
- [Techniques of Integration](https://www.vaia.com/en-us/explanations/math/calculus/techniques-of-integration/)
- [The Fundamental Theorem of Calculus](https://www.vaia.com/en-us/explanations/math/calculus/the-fundamental-theorem-of-calculus/)
- [The Limit Does Not Exist](https://www.vaia.com/en-us/explanations/math/calculus/the-limit-does-not-exist/)
- [The Mean Value Theorem](https://www.vaia.com/en-us/explanations/math/calculus/the-mean-value-theorem/)
- [The Power Rule](https://www.vaia.com/en-us/explanations/math/calculus/the-power-rule/)
- [The Squeeze Theorem](https://www.vaia.com/en-us/explanations/math/calculus/the-squeeze-theorem/)
- [The Trapezoidal Rule](https://www.vaia.com/en-us/explanations/math/calculus/the-trapezoidal-rule/)
- [Theorems of Continuity](https://www.vaia.com/en-us/explanations/math/calculus/theorems-of-continuity/)
- [Topology](https://www.vaia.com/en-us/explanations/math/calculus/topology/)
- [Trigonometric Substitution](https://www.vaia.com/en-us/explanations/math/calculus/trigonometric-substitution/)
- [Triple Integral](https://www.vaia.com/en-us/explanations/math/calculus/triple-integral/)
- [Triple Integral Spherical Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/triple-integral-spherical-coordinates/)
- [Triple Integrals in Cylindrical Coordinates](https://www.vaia.com/en-us/explanations/math/calculus/triple-integrals-in-cylindrical-coordinates/)
- [Types of Discontinuity](https://www.vaia.com/en-us/explanations/math/calculus/types-of-discontinuity/)
- [Using Slope Fields to Graph Solutions](https://www.vaia.com/en-us/explanations/math/calculus/using-slope-fields-to-graph-solutions/)
- [Variational Methods](https://www.vaia.com/en-us/explanations/math/calculus/variational-methods/)
- [Vector Valued Function](https://www.vaia.com/en-us/explanations/math/calculus/vector-valued-function/)
- [Vectors In Space](https://www.vaia.com/en-us/explanations/math/calculus/vectors-in-space/)
- [Vectors in Calculus](https://www.vaia.com/en-us/explanations/math/calculus/vectors-in-calculus/)
- [Velocity as Average Rate of Change](https://www.vaia.com/en-us/explanations/math/calculus/velocity-as-average-rate-of-change/)
- [Vertical Asymptote](https://www.vaia.com/en-us/explanations/math/calculus/vertical-asymptote/)
- [Volume Integrals](https://www.vaia.com/en-us/explanations/math/calculus/volume-integrals/)
- [Volume by disks](https://www.vaia.com/en-us/explanations/math/calculus/volume-by-disks/)
- [Volume by shells](https://www.vaia.com/en-us/explanations/math/calculus/volume-by-shells/)
- [Washer Method](https://www.vaia.com/en-us/explanations/math/calculus/washer-method/)
- [Decision Maths](https://www.vaia.com/en-us/explanations/math/decision-maths/)
- [Discrete Mathematics](https://www.vaia.com/en-us/explanations/math/discrete-mathematics/)
- [Geometry](https://www.vaia.com/en-us/explanations/math/geometry/)
- [Logic and Functions](https://www.vaia.com/en-us/explanations/math/logic-and-functions/)
- [Mechanics Maths](https://www.vaia.com/en-us/explanations/math/mechanics-maths/)
- [Probability and Statistics](https://www.vaia.com/en-us/explanations/math/probability-and-statistics/)
- [Pure Maths](https://www.vaia.com/en-us/explanations/math/pure-maths/)
- [Statistics](https://www.vaia.com/en-us/explanations/math/statistics/)
- [Theoretical and Mathematical Physics](https://www.vaia.com/en-us/explanations/math/theoretical-and-mathematical-physics/)
Contents
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- Last Updated: 04.04.2023
- 9 min reading time
- Content creation process designed by
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- [Hydrostatic Pressure Definition](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-0)
- [Hydrostatic Pressure Formula](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-1)
- [Hydrostatic Force](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-2)
- [Hydrostatic Force Formula](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-3)
- [Hydrostatic Forces on Submerged Surfaces](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-4)
- [Hydrostatic Pressure - Key takeaways](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-5)
- [Similar topics in Math](https://www.vaia.com/en-us/explanations/math/calculus/hydrostatic-pressure/#anchor-6)
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## Hydrostatic Pressure Definition
So what is hydrostatic pressure?
**Hydrostatic pressure** is the pressure exerted by a fluid at rest due to the force of gravity.
Hydrostatic pressure varies depending on the fluid and the depth.
## Hydrostatic Pressure Formula
You will need some variables going forward:
- P is the hydrostatic pressure in N / m 2 ;
- ρ is the density of the fluid in kg / m 3 ;
- g is the acceleration due to gravity in m / s 2; and
- h is the height of the water column in m.
Then the formula for hydrostatic pressure is
P \= ρ g h .
Notice that pressure increases as the height of the water column (also called the depth) increases. That fact is illustrated in the image below.
 Fig. 2: The hydrostatic pressure, ranked from lowest to highest, would go A , B , C .
The density of water is 1000 kg / m 3, and for gravity on Earth you can use 9\.81 m / s 2.
## Hydrostatic Force
You are probably wondering what the difference between hydrostatic force and pressure is.
**Hydrostatic force** is the force caused by the pressure of a fluid acting on a submerged surface.
If you think about the example of the diver at the start of the article, the hydrostatic force on the diver is the force of the water on their submerged body. As you have seen, pressure increases as the height of the water column increases. In other words, the deeper the diver goes, the higher the pressure is, which in turn means the hydrostatic force acting on their body increases as well.
## Hydrostatic Force Formula
The hydrostatic force formula is
F \= P A ,
where:
- F is the hydrostatic force in N;
- P is they hydrostatic pressure in N / m 2; and
- A is the area of the submerged surface in m 2 .
Notice that the formula works for very complicated surfaces, like that of a diver's body. However figuring out the exact hydrostatic force on a diver is a bit of a big problem to start with, so instead let's look at submerging a flat plate in a fluid instead.
 Fig. 3: A flat plate submerged d units under the water.
In the picture above, there is a flat hexagonal plate submerged d units under the water. The rectangle shown has height Δ y and width x i, so the area of the rectangle is x i Δ y. That means the hydrostatic force on the rectangle is
F rectangle \= P A \= ( ρ g h ) ( x i Δ y ) \= ( 1000 ⋅ 9\.81 d i ) ( x i Δ y ) \= 9810 d i x i Δ y N .
An approximation of the hydrostatic force on the entire plate would be the sum of the hydrostatic forces on all of the rectangles, so if there are n rectangles,
F plate ≈ ∑ i \= i n 9810 d i x i Δ y .
This should look familiar to you since finding areas this way is one of the applications of integrals\!
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## Hydrostatic Forces on Submerged Surfaces
Let's continue on with the plate example started in the previous section. The first step is to make a properly labeled diagram with the x and y\-axis included. This plate is vertically symmetric, so it makes sense to center it on the y\-axis with the bottom of the plate on the x axis.
The top of the plate is 6 meters below the surface of the water. The plate can be thought of in two sections: the upper section where y ≥ 2; and the lower section where y ≤ 2.
Fig. 4: The plate placed on the coordinate axis with the depth and equation of the upper line of the plate labeled.
Then the total area of the plate is the sum of the upper section area and the lower section area. Therefore when integrating you can split the integral at y \= 2 to make calculations easier. In fact, since the area is symmetric across the y axis, you can find the area of the right half of the plate and then multiply by 2 to get the total area of the plate.
### Upper Plate Calculations
For the upper part of the plate, the depth below the surface is d i. The equation of the line describing the edge of the plate is y \= − 0\.8 x \+ 8.
The summation for the hydrostatic force on the upper part of the plate is
F upper ≈ ∑ i \= i n 9810 d i x i Δ y ,
which has a Δ y in it. This means the integration will be done with respect to y and not with respect to x. Therefore solving the equation of the line for x in terms of y gives
x \= − 1\.25 y \+ 10\.
You will need to be careful since writing it this way only gives you the right half of the plate! Since the plate is symmetric, to find the area from the left side of the plate to the right side of the plate, you can multiply the summation by 2. Substituting the line equation into the summation and multiplying by 2 gives
F upper ≈ 2 ∑ i \= i n 9810 d i ( − 1\.25 y i \+ 10 ) Δ y .
That just leaves writing d i in terms of y i. As you can see from the picture above,
y i \= 12 − d i ,
and now the summation becomes
F upper ≈ 2 ∑ i \= i n 9810 ( 12 − y i ) ( − 1\.25 y i \+ 10 ) Δ y .
Then the total force on the upper half of the plate is
F upper \= 2 lim n → ∞ ∑ i \= 1 n 9810 ( 12 − y i ) ( − 1\.25 y i \+ 10 ) Δ y \= 2 ∫ 2 6 9810 ( 12 − y ) ( − 1\.25 y \+ 10 ) d y \= 2 ( 9810 ) ( 5 y 3 − 150 y 2 \+ 1440 y 12 ) \| y \= 2 y \= 6 \= 2 ( 9810 ) ( 500 3 ) \= 3270000 N .
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### Lower Plate Calculations
Let's look at the lower plate. The diagram below shows the equation of the line for that part of the plate.
Fig. 5: Lower half of the submerged plate.
The equation of the line is y \= 0\.4 x − 1. As in the calculations for the upper half of the plate, you need to write this equation in terms of x so it can be substituted in. Rewriting the equation of the line in terms of x you get x \= 2\.5 y \+ 2\.5. Since d i \= 12 − y i , the hydrostatic force on a horizontal rectangular strip running from the y\-axis to the line is
F i \= P i A i \= 9810 d i ( 2\.5 y i \+ 2\.5 ) Δ y \= 9810 ( 12 − y i ) ( 2\.5 y i \+ 2\.5 ) Δ y .
The total force on the upper half of the plate is then
F lower \= 2 lim n → ∞ ∑ i \= 1 n 9810 ( 12 − y i ) ( 2\.5 y i \+ 2\.5 ) Δ y \= 2 ( 2\.5 ) ( 9810 ) ∫ 0 2 ( 12 − y ) ( y \+ 1 ) d y \= ( 49050 ) − 2 y 3 \+ 33 y 2 \+ 72 y 6 \| y \= 0 y \= 2 \= ( 49050 ) ( 130 3 ) \= 2125500 N .
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
### Total Hydrostatic Force on the Plate
Then to get the total hydrostatic force on the plate, you just need to add the hydrostatic force on the upper part of the plate to the hydrostatic force on the lower part of the plate:
F total \= F upper \+ F lower \= 3270000 \+ 2125500 \= 5395500 N .
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
## Hydrostatic Pressure - Key takeaways
- Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity.
- Then the formula for hydrostatic pressure is
P \= ρ g h .where:
- P is the hydrostatic pressure in N / m 2 ;
- ρ is the density of the fluid in kg / m 3 ;
- g is the acceleration due to gravity in m / s 2; and
- h is the height of the water column in m.
- Hydrostatic force is the force caused by the pressure of a fluid acting on a submerged surface.
- The hydrostatic force formula is
F \= P A ,
where:
- F is the hydrostatic force in N;
- P is they hydrostatic pressure in N / m 2; and
- A is the area of the submerged surface in m 2 .
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[If you submerged two flat plates, one a square and one a circle, to the same depth, and the diameter of the circle is the same as the length of the side of the square, which would have lower hydrostatic pressure? The circle.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
[Which of the following would increase the hydrostatic force? Increasing the area of the submerged object.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
[Which of the following would increase the hydrostatic pressure? Having a higher gravity.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
[Suppose you have a water column and you are measuring the hydrostatic force at three different heights: 1 meter, 2 meters, and 3 meters. Which measurement will give the lowest hydrostatic force? The measurement at 1 meter.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
[Suppose that both a professional football player and a small child went diving underwater to the same depth. Which one would experience greater hydrostatic force? The football player.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
[Suppose you have a water column and you are measuring the hydrostatic force at three different heights: 1 meter, 2 meters, and 3 meters. Which measurement will give the greatest hydrostatic force? The measurement at 3 meters.](https://app.vaia.com/signup-website/?lang=en&web_campaign=schule_math_calculus_hydrostatic-pressure&is_pupil=true&maintopic_id=21605743&summary_id=40563452&course_subject=6648097&web_source=website&_smtype=3&domain=com&conversion_element=flashcards-list-body-link&target=flashcard-list)
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## Frequently Asked Questions about Hydrostatic Pressure
What is hydrostatic pressure?
Hydrostatic pressure is the pressure exerted by a column of fluid.
What generates the force that results in hydrostatic pressure?
Gravity.
How to calculate hydrostatic pressure?
It depends on the depth in the fluid the object is submerged and the shape of the object. But for any point in a water column, to calculate the pressure we use:
P=ρgh
Here h is the depth, g is the gravity and is the density ρ of the fluid.
What is the basic function of hydrostatic pressure?
*P=ρgd*, where *P* is pressure, *ρ* is the density of the fluid, *g* is gravitational acceleration, and *d* or *h* is the depth below the surface.
What is the hydrostatic forces?
The hydrostatic force that acts on an area is *F=PA*, where *F* is force, *P* is hydrostatic pressure, and *A* is the area of the object.
How to measure hydrostatic pressure?
Usually with a gauge submerged in the fluid which returns the force per unit area.
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| Readable Markdown | - Fact Checked Content
- Last Updated: 04.04.2023
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Fig. 1: Stay here and learn more to keep your brain from getting the metaphorical bends\!
Test your knowledge with multiple choice flashcards
## Hydrostatic Pressure Definition
So what is hydrostatic pressure?
**Hydrostatic pressure** is the pressure exerted by a fluid at rest due to the force of gravity.
Hydrostatic pressure varies depending on the fluid and the depth.
## Hydrostatic Pressure Formula
You will need some variables going forward:
- P is the hydrostatic pressure in N / m 2 ;
- ρ is the density of the fluid in kg / m 3 ;
- g is the acceleration due to gravity in m / s 2; and
- h is the height of the water column in m.
Then the formula for hydrostatic pressure is
P \= ρ g h .
Notice that pressure increases as the height of the water column (also called the depth) increases. That fact is illustrated in the image below.
 Fig. 2: The hydrostatic pressure, ranked from lowest to highest, would go A , B , C .
The density of water is 1000 kg / m 3, and for gravity on Earth you can use 9\.81 m / s 2.
## Hydrostatic Force
You are probably wondering what the difference between hydrostatic force and pressure is.
**Hydrostatic force** is the force caused by the pressure of a fluid acting on a submerged surface.
If you think about the example of the diver at the start of the article, the hydrostatic force on the diver is the force of the water on their submerged body. As you have seen, pressure increases as the height of the water column increases. In other words, the deeper the diver goes, the higher the pressure is, which in turn means the hydrostatic force acting on their body increases as well.
## Hydrostatic Force Formula
The hydrostatic force formula is
F \= P A ,
where:
- F is the hydrostatic force in N;
- P is they hydrostatic pressure in N / m 2; and
- A is the area of the submerged surface in m 2 .
Notice that the formula works for very complicated surfaces, like that of a diver's body. However figuring out the exact hydrostatic force on a diver is a bit of a big problem to start with, so instead let's look at submerging a flat plate in a fluid instead.
 Fig. 3: A flat plate submerged d units under the water.
In the picture above, there is a flat hexagonal plate submerged d units under the water. The rectangle shown has height Δ y and width x i, so the area of the rectangle is x i Δ y. That means the hydrostatic force on the rectangle is
F rectangle \= P A \= ( ρ g h ) ( x i Δ y ) \= ( 1000 ⋅ 9\.81 d i ) ( x i Δ y ) \= 9810 d i x i Δ y N .
An approximation of the hydrostatic force on the entire plate would be the sum of the hydrostatic forces on all of the rectangles, so if there are n rectangles,
F plate ≈ ∑ i \= i n 9810 d i x i Δ y .
This should look familiar to you since finding areas this way is one of the applications of integrals\!
## Hydrostatic Forces on Submerged Surfaces
Let's continue on with the plate example started in the previous section. The first step is to make a properly labeled diagram with the x and y\-axis included. This plate is vertically symmetric, so it makes sense to center it on the y\-axis with the bottom of the plate on the x axis.
The top of the plate is 6 meters below the surface of the water. The plate can be thought of in two sections: the upper section where y ≥ 2; and the lower section where y ≤ 2.
Fig. 4: The plate placed on the coordinate axis with the depth and equation of the upper line of the plate labeled.
Then the total area of the plate is the sum of the upper section area and the lower section area. Therefore when integrating you can split the integral at y \= 2 to make calculations easier. In fact, since the area is symmetric across the y axis, you can find the area of the right half of the plate and then multiply by 2 to get the total area of the plate.
### Upper Plate Calculations
For the upper part of the plate, the depth below the surface is d i. The equation of the line describing the edge of the plate is y \= − 0\.8 x \+ 8.
The summation for the hydrostatic force on the upper part of the plate is
F upper ≈ ∑ i \= i n 9810 d i x i Δ y ,
which has a Δ y in it. This means the integration will be done with respect to y and not with respect to x. Therefore solving the equation of the line for x in terms of y gives
x \= − 1\.25 y \+ 10\.
You will need to be careful since writing it this way only gives you the right half of the plate! Since the plate is symmetric, to find the area from the left side of the plate to the right side of the plate, you can multiply the summation by 2. Substituting the line equation into the summation and multiplying by 2 gives
F upper ≈ 2 ∑ i \= i n 9810 d i ( − 1\.25 y i \+ 10 ) Δ y .
That just leaves writing d i in terms of y i. As you can see from the picture above,
y i \= 12 − d i ,
and now the summation becomes
F upper ≈ 2 ∑ i \= i n 9810 ( 12 − y i ) ( − 1\.25 y i \+ 10 ) Δ y .
Then the total force on the upper half of the plate is
F upper \= 2 lim n → ∞ ∑ i \= 1 n 9810 ( 12 − y i ) ( − 1\.25 y i \+ 10 ) Δ y \= 2 ∫ 2 6 9810 ( 12 − y ) ( − 1\.25 y \+ 10 ) d y \= 2 ( 9810 ) ( 5 y 3 − 150 y 2 \+ 1440 y 12 ) \| y \= 2 y \= 6 \= 2 ( 9810 ) ( 500 3 ) \= 3270000 N .
### Lower Plate Calculations
Let's look at the lower plate. The diagram below shows the equation of the line for that part of the plate.
Fig. 5: Lower half of the submerged plate.
The equation of the line is y \= 0\.4 x − 1. As in the calculations for the upper half of the plate, you need to write this equation in terms of x so it can be substituted in. Rewriting the equation of the line in terms of x you get x \= 2\.5 y \+ 2\.5. Since d i \= 12 − y i , the hydrostatic force on a horizontal rectangular strip running from the y\-axis to the line is
F i \= P i A i \= 9810 d i ( 2\.5 y i \+ 2\.5 ) Δ y \= 9810 ( 12 − y i ) ( 2\.5 y i \+ 2\.5 ) Δ y .
The total force on the upper half of the plate is then
F lower \= 2 lim n → ∞ ∑ i \= 1 n 9810 ( 12 − y i ) ( 2\.5 y i \+ 2\.5 ) Δ y \= 2 ( 2\.5 ) ( 9810 ) ∫ 0 2 ( 12 − y ) ( y \+ 1 ) d y \= ( 49050 ) − 2 y 3 \+ 33 y 2 \+ 72 y 6 \| y \= 0 y \= 2 \= ( 49050 ) ( 130 3 ) \= 2125500 N .
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
### Total Hydrostatic Force on the Plate
Then to get the total hydrostatic force on the plate, you just need to add the hydrostatic force on the upper part of the plate to the hydrostatic force on the lower part of the plate:
F total \= F upper \+ F lower \= 3270000 \+ 2125500 \= 5395500 N .
As you would expect, the hydrostatic force on the lower part of the plate is smaller than that on the upper part of the plate because the area on the lower plate is smaller than the area on the upper part of the plate.
## Hydrostatic Pressure - Key takeaways
- Hydrostatic pressure is the pressure exerted by a fluid at rest due to the force of gravity.
- Then the formula for hydrostatic pressure is
P \= ρ g h .where:
- P is the hydrostatic pressure in N / m 2 ;
- ρ is the density of the fluid in kg / m 3 ;
- g is the acceleration due to gravity in m / s 2; and
- h is the height of the water column in m.
- Hydrostatic force is the force caused by the pressure of a fluid acting on a submerged surface.
- The hydrostatic force formula is
F \= P A ,
where:
- F is the hydrostatic force in N;
- P is they hydrostatic pressure in N / m 2; and
- A is the area of the submerged surface in m 2 .
## Frequently Asked Questions about Hydrostatic Pressure
What is hydrostatic pressure?
Hydrostatic pressure is the pressure exerted by a column of fluid.
What generates the force that results in hydrostatic pressure?
Gravity.
How to calculate hydrostatic pressure?
It depends on the depth in the fluid the object is submerged and the shape of the object. But for any point in a water column, to calculate the pressure we use:
P=ρgh
Here h is the depth, g is the gravity and is the density ρ of the fluid.
What is the basic function of hydrostatic pressure?
*P=ρgd*, where *P* is pressure, *ρ* is the density of the fluid, *g* is gravitational acceleration, and *d* or *h* is the depth below the surface.
What is the hydrostatic forces?
The hydrostatic force that acts on an area is *F=PA*, where *F* is force, *P* is hydrostatic pressure, and *A* is the area of the object.
How to measure hydrostatic pressure?
Usually with a gauge submerged in the fluid which returns the force per unit area.
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